On the Spectral Theory of Operator Pencils in a Hilbert Space

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Journal of Nonlinear Mathematical Phsics ISSN: 1402-9251 Print 1776-0852 Online

com/loi/tnmp20 On the Spectral Theor of Operator Pencils in a Hilbert Space Roman

Andrushkiw 1995 On the Spectral Theor of Operator Pencils in a Hilbert Space,

2.3-4.15 To link to this article: https://doi.org/10.2991/jnmp.1995.2.3-4.15 Published online: 21 Jan 2013.

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1 Journal of Nonlinear Mathematical Phsics ISSN: Print Online Journal homepage: On the Spectral Theor of Operator Pencils in a Hilbert Space Roman I. Andrushkiw To cite this article: Roman I. Andrushkiw 1995 On the Spectral Theor of Operator Pencils in a Hilbert Space, Journal of Nonlinear Mathematical Phsics, 2:3-4, , DOI: / jnmp To link to this article: Published online: 21 Jan Submit our article to this journal Article views: 79 View related articles Citing articles: 1 View citing articles Full Terms & Conditions of access and use can be found at Download b: [ ] Date: 25 November 2017, At: 12:00

2 Nonlinear Mathematical Phsics 1995, V.2, N 3 4, On the Spectral Theor of Operator Pencils in a Hilbert Space Roman I. ANDRUSHKIW Department of Mathematics, Center for Applied Mathematics and Statistics, New Jerse Institute of Technolog, Newark, New Jerse Abstract Consider the operator pencil L λ = A λb λ 2 C, where A, B, and C are linear, in general unbounded and nonsmmetric, operators densel defined in a Hilbert space H. Sufficient conditions for the eistence of the eigenvalues of L λ are investigated in the case when A, B and C are K-positive and K-smmetric operators in H, and a method to bracket the eigenvalues of L λ is developed b using a variational characterization of the problem i L λ u = 0. The method generates a sequence of lower and upper bounds converging to the eigenvalues of L λ and can be considered an etension of the Temple-Lehman method to quadratic eigenvalue problems i. 1 Introduction Let H be a separable comple Hilbert space with the inner product and norm and consider in H the nonlinear eigenvalue problem,, =, 1/2,, H 1 A λb λ 2 C = 0 2 where A and C are K-p.d. operators with D C D A, D A is dense in H, and B is an operator with D B D C. Recall [1 3] that b the definition of A and C there eists a closable operator K with D K D C mapping D A onto a dense subset KD A of H and positive constants α 1, α 2, β 1, β 2 such that A, K α 1 2, D A 3 K 2 α 2 A, K, D A 4 C, K β 1 2, D C 5 K 2 β 2 C, K, D C 6 The class of K-p.d. operators {P } contains, among others, the following families of mappings: Copright c 1995 b Mathematical Ukraina Publisher. All rights of reproduction in an form reserved.

3 ON THE SPECTRAL THEORY OF OPERATOR PENCILS 357 a Positive definite operators; in this case K is the identit map or, if P is also self-adjoint, K can be an root of P. b Closeable and densel invertible 1 operators; in this case we let K = P. c The operator P of the form P = S 2j+1 or P = S 2j+2 where for some i, 0 i < j, the operator S 2j+i+1 is positive definite; in this case we let K = S 2i+1 or K = S 2i+2, provided that K so defined is closable and KD P is dense in H. To this class belong, in particular, ordinar differential operators of odd and even order and weakl elliptic partial differential operators of odd and even order which in general are not self-adjoint [2]. d A subclass of bounded and unbounded smmetrizable operators investigated b a number of authors [2, 4]. Let D[A] be the set D A endowed with the new metric, A = A, K, 2 A =, A,, D A 7 and denote b H A the completion of D[A] in the metric 7. Similarl, let D[C] be the set D C with the metric, C = C, K, 2 C =, C,, D C 8 and define H C to be the completion of D[C] in the metric 8. One can show that the space H A is contained in H in the sense of uniquel identifing the elements of H A with certain elements in H and clearl, since C is K-p.d., the above assertion is valid also for the space H C, i.e., H C H. Let H 1 = H H C be the Cartesian product space, with the norm and inner product defined b u = u, v 1 =, p +, q C and v = p q H H C 9 u 1 = u, u 1/2 1/2 1 = 2 + C Clearl, H 1 is a Hilbert space and, since H C is a subset of H, it follows that H 1 H H in the sense mentioned above. Now, let T : D A D[C] H 1 H 1 be the operator matri T = A 0 0 I, T = A, u = D A D[C]. 11 Similarl, let us define in H 1 the operators S = B C I 0 ˆK = K 0 0 I, S : D B D[C] H 1 H 1, 12, ˆK : DK D[C] H 1 H An operator P will be called invertible if it has a bounded inverse, densel invertible if it is invertible and its range R P is dense in H, and continuousl invertible if it is densel invertible and R P = H.

4 358 R. ANDRUSHKIW Observe that the quadratic eigenvalue problem 2 is equivalent to the sstem A λb λc = 0 λ = 0 which, in view of 11 and 12, is equivalent to the linear equation 14 T u λsu = 0 15 in the sense that if i is a solution of 2 corresponding to λ = λ i, then u i = i i = λ i i is a solution of 15 and, conversel, if u i is a solution of 15 corresponding to λ = λ i, then i = λ i i and i is a solution of 2. Proposition 1. The operator T defined b 11 is ˆK-p.d. in the space H 1 = H H C ; i.e., T satisfies the following conditions: a D T is dense in H 1. b D ˆK D T and ˆKD T is dense in H 1. c ˆK is closable in H 1. d There eist positive constants γ 1, γ 2 such that i with T u, ˆKu 1 γ 1 u 2 1, u D T, 16 ˆKu 2 1 γ 2 T u, ˆKu 1, u D T. 17 Proof. a Let u = be an arbitrar element in H 1 = H H C. Since D A is dense in H, there eists a sequence { n } D A which converges to in the H-metric. Similarl, since D C is dense in H C, there eists a sequence { n } D C which converges to in the n H C -metric. Hence, if we define a sequence in D T = D A D[C] b u n =, then n lim u n u 2 n 1 = lim n 2 + n 2 n C = 0. b B definition, ˆKDT = KD A D[C] where KD A is dense in H. Hence, using a similar argument as in part a, one can show that ˆKD T is dense in H 1. Moreover, since D K D A, it follows that c Let u n = n n that the following conditions hold: D ˆK = D K D[C] D A D[C] = D T. 18 be a sequence in D ˆK, and f = f1 f 2 an element in H 1 such lim u n 1 = 0, 19 n lim ˆKu n f 1 = n From 20 we obtain lim Kn f n f 2 2 n c = 0 which implies that K n f 1 H, 21

5 ON THE SPECTRAL THEORY OF OPERATOR PENCILS 359 n f 2 H C, 22 On the other hand, from 19 we deduce that n 0 in the H-norm, and 23 n 0 in the H C -norm. 24 In view of 21 and 23 it follows that f 1 = 0, since K is closable in H. Moreover, from f1 22 and 24 it follows that f 2 = 0. Hence, f = = 0, and f ˆK is closable in H 1. 2 d If u = D T, then T u, ˆKu 1 = A, K +, C and, in view of 3, we obtain the inequalit T u, ˆKu 1 α C. 25 Let γ 1 = min{α 1, 1}; then from 25 and 10 it follows that T u, ˆKu 1 γ 1 u 2 1, u D T. 26 Since ˆKu 2 1 = K, K +, C, it follows from 4 and 11 that where γ 2 = ma{α 2, 1}. ˆKu 2 1 α 2 A, K +, C γ 2 T u, ˆKu 1, u D T, 27 p Let u = and v = be elements of the space D q T = D A D[C] H H C = H 1 and let us introduce in D T a new norm and inner product u, v 2 = T u, ˆKv 1 =, p A +, q C 28 u 2 = 2 A + 2 C. 29 Define b D[T ] the linear set D T endowed with the metric 2 2 =, 2, and observe that D[T ] = D[A] D[C]. 30 In view of 16, 17 and the fact that T is K-p.d. in H 1, we have the inequalities u 2 γ 1 u 1, u D T 31 ˆKu 1 γ 2 u 2, u D T 32 Clearl, D T satisfies all the properties of a Hilbert space, with the possible eception of completeness. Let us denote b H 2 the completion of D[T ] in the metric 29. Proposition 2. a H 2 = H A H C. b H 2 is contained in H 1 in the sense of identifing uniquel the elements from H 2 with certain elements in H 1.

6 360 R. ANDRUSHKIW c ˆK can be etended to a bounded operator ˆK 0 mapping all of H 2 to H 1 such that ˆK ˆK 0 ˆK, where ˆK denotes the closure of ˆK in H 1. d T has a unique closed ˆK 0 -p.d. etension T 0 such that T 0 T, T 0 has a bounded inverse T0 1 defined on all of H 1 = R T0, and the inequalities 31 and 32 remain valid in H 2 in the form u 2 γ 1 u 1, u H 2 33 ˆK 0 u 1 γ 2 u 2, u H Proof. The proof of part a follows from 30 and the fact that H A and H C are the completions of the spaces D[A] and D[C] in the norms A and C, respectivel. B Proposition 1, the operator T is ˆK-p.d. in H 1. Hence, the proof of parts b, c, and d can be derived from Lemma 1.2 of Petrshn [3], provided the spaces H 2, H 1 and the operator ˆK in Proposition 2 are identified with H 0, H, and K, in Lemma 1.2, respectivel. In the sequel we shall assume, when necessar, that the operators ˆK and T have alread been etended and the notation T 0 and ˆK 0 will not be used. Note that in applications it is often not necessar to etend the operators T and ˆK. 2 The equaivalent linear problem T u λsu = 0 Definition 1. The quadratic eigenvalue problem A λb λ 2 C = 0, 35 where A and C are K-p.d. with D A D C D B and B is K-smmetrric on D C, i.e., will be called K-real. B, K = K, B,, D C 36 Proposition 3. If the quadratic eigenvalue problem 35 is K-real in H, then the equivalent linear problem T u λsu = 0 37 defined b is ˆK-real in H 1 = H H C, i.e. T is ˆK-p.d. and S is ˆK-smmetric on D T. Proof. In view of Proposition 1, onl the ˆK-smmetr of S needs to be verified. To this p end let u = and v = be elements in D q T H 1 and note that Su, ˆKv 1 = B + C, Kp +, q C = B, Kp + C, Kp + C, Kq 38 Since b definition the operators B and C are K-smmetric on D A equation ields the identit H, the above Su, ˆKv 1 = K, Bp + Cq + C, Kp = ˆKu, Sv 1, u, v D T 39 which proves the ˆK-smmetr of S on D T.

7 ON THE SPECTRAL THEORY OF OPERATOR PENCILS 361 Let us assume that the eigenvalue problem 35 is K-real, which implies that problem 37 is ˆK-real. A value of the comple parameter λ for which 37 has a nontrivial solution u D T will be called an eigenvalue of 37, and u its corresponding eigenfunction. The set of all eigenvalues of 37 will be denoted b pσ37 and called the point spectrum of 37. B the multiplicit of λ we shall mean the number of linearl independent eigenfunctions which correspond to λ. Since T, S are ˆK-smmetric, it follows [5] that the eigenvalues of 37 are real, and the eigenfunctions u 1, u 2 corresponding to distinct eigenvalues λ 1, λ 2 are orthogonal in the sense that T u 1, ˆKu 2 1 = 0. Since the space H 1 is separable, it follows that the point spectrum of 37 is countable. Suppose the operators K and L λ A λb λ 2 C are closed with D K = D C, and that L λ : D A H H is a bijection for all λ, ecept possibl for a discrete set of eigenvalues of the problem A λb λ 2 C = 0. Under the above assumptions, it is not difficult to show that the equivalent linear problem T u λsu = 0 in H 1 = H H C satisfies the following conditions: α: The operator G λ = T λs : D T H 1 H 1 is continuousl invertible for all λ / pσ 37. β: The spectrum σn of the operator N = T 1 S : D[T ] H 2 D[T ], contains onl eigenvalues of finite multiplicit with zero as its sole possible limit point. Let pσl λ = {λ i : i = 1, 2,..} denote the point spectrum of the operator L λ, with the eigenvalues ordered according to increasing magnitude and repeated as man times as their multiplicit indicates. Let { i : i = 1, 2,...} be the set of corresponding eigenfunctions, normalized in the sense that i 2 A + λ2 i i 2 C = 1. Then, using certain results from the theor of linear K-real eigenproblem T u λsu = 0, we ma derive the following theorems, which etend the corresponding results [6 8] obtained for the case when C is the identit operator and A, B are self-adjoint, positive definite, or compact operators. Related results, under different assumptions on the operators A,B,C, have been obtained b other authors [9 15]. Theorem 1. Assume that the eigenproblem 35 is K-real, that L λ : D A H is a bijection for all λ / pσl λ and that the operators L λ and K are closed with D K = D C. Then the eigenvalues and eigenfunctions of problem 35 have the variational characterization 1 λ n = sup, T D A D C { E, :, i A + λ i, i C = 0, 1 i n 1} = E n, λ n n, 40 B, K + 2ReC, K where E, =. A, K + C, K Moreover, the eigenvalues found b this variational process ehaust entirel the set pσl λ. Proof. B hpothesis the linearized eigenproblem 37 is ˆK-real and satisfies conditions α and β. It follows from the theor of linear K-real eigenproblems [3, 5] that the eigenpairs λ i, u i of problem 37, normalized in the sense u i 2 = 1, satisf the variational principle { } 1 Su, λ n = sup ˆKu1 u D T T u, ˆKu : T u, ˆKu i 1 = 0, 1 i n 1 = 1

8 362 R. ANDRUSHKIW Su n, ˆKu n 1 / T u, ˆKu n 1 40a and the eigenvalues determined b 40a ehaust entirel the set pσ37. Thus, the validit of the last assertion of Theorem 1 follows from the fact that pσ37 = pσl λ. If we let u =, T, u DT = D A D C, then epanding the inner products in 40a and using the K-smmetr propert of the operator C, we obtain the epressions Su, ˆKu 1 = B, K + C, K + C, K = B, K + 2ReC, K T u, ˆKu 1 = A, K + C, K u i 2 2 = i 2 A + λ 2 i i 2 C T u, ˆKu i 1 =, i A + λ i, i C. Substituting the above into 40a ields the variational formula 40. Lemma 1. Assume the hpothesis of Theorem 1. a Suppose S and S + are ˆK-smmetric operators, T is ˆK-p.d., and S + u, ˆKu Su, ˆKu for u D T. Then the eigenvalues λ + i and λ i of the corresponding eigenproblems T u λ + S + u = 0 and T u λsu = 0 satisf the inequalit λ + i λ i, i=i,2,... b Suppose that T and T are ˆK-p.d. operators with D T = DT, S is K-smmetric on D T, and T u, ˆKu T u, ˆKu for u D T. Then the eigenvalues λ i and λ i of the corresponding eigenproblems T u λsu = 0 and T u λ Su = 0 satisf the inequalit λ i λ i,,2,... Proof. The proof of parts a and b is a direct consequence of the variational principle 40 in Theorem 1. Theorem 2. Assume the hpothesis of Theorem 1 and let {u i : 1, 2,...} be the set of eigenfunctions, orthonormal in H 2, of the ˆK-real eigenproblem 37. If u D T, then T 1 Su has the epansion T 1 Su = Su, ˆKu i 1 u i 41 which converges in the H 1 and H 2 -norm. Proof. The result follows directl from the corresponding eigenfunction epansion theorem [3, 5] for linear K-real eigenvalue problems. 3 Iterative method Let f 0 = 0 0 denote b f k = be an element in D T such that f 0 / NS the null space of S, and k k the iterant at the k-th step of our process; then the succeeding iterant f k+1 is obtained b solving the equation T f k+1 = Sf k, i.e., A 0 k+1 B C k =, k I I 0 k+1 k

9 ON THE SPECTRAL THEORY OF OPERATOR PENCILS 363 Now, let us determine the constants a k = Sf k i, ˆKf i 1 = B k i, K i + C k i, K i + C k i, K i 0 i k, k = 1, 2, Note that the values of Sf k i, ˆKf i 1 depend on k but not on i, since from the ˆK-smmetr of S and T it follows that Sf k, ˆKf 0 1 = Sf k 1, ˆKf 1 1 = = Sf 0, ˆKf k 1. Also, note that the elements of the sequence {f k } cannot vanish, since f 0 / NS implies that f n / NS for n 0. Indeed, if Sf n 0 for n < k, then f k = T 1 Sf k 1 0, and from the identit Sf k, ˆKf k 1 1 = ˆKf k, Sf k 1 1 = T f k, ˆKf k 1 > 0 it follows that Sf k 0. Thus, b induction, it follows that f n / NS for all n 0. Let H i 2 be the space spanned b the eigenfunction u i and denote b H i 2 the orthogonal complement of H i 2 in H 2. Proposition 4. Let c i = f 0, u i 2, i = 1, 2,... be the Fourier coefficients of f 0 with respect to the orthonormal set of eigenfunctions {u i } in H 2. Then, a f k ma be represented b the following series, converging in the H 1 and H 2 metrics: f k = c i λ k i u i, k = 0, 1, b the constants a k, determined b 43, are of the form a k = c i 2 λ k+1 i, k = 0, 1, Proof. a Appling Theorem 2 we ma epress f k in the form f k = T 1 Sf k 1 = Sf k 1, ˆKu i 1 u i, k = 1, 2, where the series converges in the H 1 and H 2 metrics. Now, let us show that the following identit is valid Sf k 1, ˆKu i 1 = c i λ k i, k = 1, 2, For k = 1 using the ˆK-smmetr of S and T, we obtain Sf 0, ˆKu i 1 = ˆKf o, Su i 1 = λ 1 i Suppose 47 is valid for n < k, then Sf k, ˆKu i 1 = λ 1 i ˆKf 0, T u i 1 = λ 1 i T f 0, ˆKu i 1 = c i λ 1 i. ˆKf k, Su i 1 = λ 1 i Sf k 1, ˆKu i 1 = c i λ k+1 i. Hence, identit 47 is valid b induction and substituting it into 46 completes the proof of part a. b Recall that the operator ˆK, understood in the etended sense, is a continuous mapping from H 2 into H 1 and that the series 44 is convergent in the H 1 and H 2 metrics. Thus, appling the epansion 44 to the last term in the identit a k = Sf k, ˆKf 0 1 = T f k+1, ˆKf 0 1 = ˆKf k+1, T f 0 1,

10 364 R. ANDRUSHKIW we obtain a k = c i λ k+1 i ˆKu i, T f 0 1 = c i 2 λ k+1 i, k = 0, 1, Let w k = a 2k 1 /a 2k+1 and note that b appling 45 we ma epress w k in the form w k = c i 2 λ 2k i / c i 2 λ 2k+1 i, k = 1, 2, Theorem 3. Assume the hpothesis of Theorem 1 and suppose that λ r < λ r+1 for some positive integer r. If f 0 is chosen from the space f 0 D[T ] [ r 1 Hi 2 ], f 0 / H r 2, r 1, 50 then the following statements are true: a the sequence { w k } converges monotonicall from above to λ r, b s k = λ 2k r f 2k, k = 1, 2,... converges in the H 2 -metric to an eigenfunction c r u r H r 2. Proof. a To show monotonicit of the sequence {w k }, let z k D T be defined b z k = a 2k+3 f k a 2k+1 f k+2, k = 1, 2,... Then, 0 T z k, ˆKz k 1 = a 2k+3 a 2k+3 a 2k 1 a 2 2k+1 which ields 0 a 2k 1 /a 2k+1 a 2k+1 /a 2k+3 w k w k+1, k = 1, 2,... To prove convergence, we ma use 48 to epress w k = a 2k+1 /a 2k 1 in the form w k = c i 2 λ 2k i / c i 2 λ 2k+1 i, k = 1, 2, Using the simplified notation, Λ i = λ 2 i, and the fact that b hpotesis c 1 = c 2 =... = c r 1 = 0, we deduce from 51 the epression where Pk and Qk are the series w k = Λ r P k Qk, k = 1, 2,..., 52 P k = c i 2 Λ r /Λ i k, Qk = c i 2 Λ r /Λ i k+1 i=r i=r From Bessel s inequalit c i 2 f and the fact that Λ r/λ i < 1 for i > r, it follows i=r that the series P k and Qk are uniforml convergent with respect to the parameter k, and their difference ma be epressed in the form P k Qk = c i 2 Λ r /Λ i k Λ r /Λ i k+1 c i 2 Λ r /Λ i k i=r i=r+1

11 ON THE SPECTRAL THEORY OF OPERATOR PENCILS 365 Λ r /Λ r+1 k f 0 2 2, k Since P k Qk c r 2 > 0, it follows from 53 that [P k Qk] 0 and P k/qk 1 as k. Therefore, from 52 it follows that w k converges to Λ r = λ 2 r. b B Proposition 4, Eq.44, the elements of the sequence s k = λ 2k r f 2k, k = 1, 2,... ma be represented b the series s k = λ 2k r i=r c i λ 2k i u i = c i Λ r /Λ i k u i, k = 0, 1,... i=r convergent in H 1 and H 2 -metrics. Hence, due to the orthonormalit of the eigenvectors u i in H 2, i = 1, 2,..., it follows that s k c r u r 2 2 = i=r+1 c i Λ r /Λ i k u i 2 2 = i=r+1 c i 2 Λ r /Λ i 2k. Appling Bessel s inequalit and the fact that b hpothesis Λ r < Λ r+1 Λ r+2..., we obtain the error estimate s k c r u r 2 2 f 0 2 2Λ r /Λ r+1 2k, k = 1, 2,.... Thus, it follows that the sequence {s k } converges in the H 2 -metric to an eigenfunction c r u r H2 r, with the error estimate given above. Now, let us assume that a lower bound l r+1 for the eigenvalue λ r+1 can be determined b some method such as, for eample, suggested b Lemma 1. Then, using the iterative process 43, we can derive a sequence of lower bounds that converges to λ r. Theorem 4. Assume the hpothesis of Theorem 1. If l r+1 is a lower bound for λ r+1 such that for some positive integer N we have w N l r+1 λ r+1, then Λ r l 2 r+1 w k w k+1 /l 2 r+1 w k+1 for k N and the sequence of lower bounds converges to Λ r as k. Proof. The proof of the above theorem is based on the corresponding results for linear K-real eigenvalue problems see [5], p.207. Theorems 3-4 allow us to bracket the eigenvalues of a quadratic eigenvalue problem 35 L λ = 0 b a procedure which is similar to the Temple-Lehman method for linear eigenvalue problems Mu λnu = 0. In that sense the above results ma be considered an etension of the Temple-Lehman method to nonlinear quadratic eigenvalue problems 35, where A,B,C are smmetrizable operators in H. Important etensions and applications of the Temple-Lehman method to linear problems Mu λnu = 0, where M and N are partial differential operators, ma be found in the work of F. Goerisch and H. Haunhorst [16].

12 366 R. ANDRUSHKIW References [1] Andrushkiw R. I., J. Math. Anal.Appl., 1975, V.50, [2] Filippov V.M., Variational Principles for Nonpotential Operators, AMS Trans. Math. Monog., 1989, V.77. [3] Petrshn W.V., Phil.Trans. Roal Soc. London, 1968, V.262, [4] Albrecht J., Goerisch F., Eine einheitliche Herleitung von Einschliessungssätzen für Eigenwerte, ISNM, 1984, V.69, [5] Andrushkiw R.I., Numer. Funct. Anal. Optimiz., 1983, V.6, [6] Abramov Y.S., Pencils of waveguide tpe and related etremal problems, Russian, Prob. Mat. Anal., 1990, V.11, [7] Tarnai T., Acta Technica Acad. Scient. Hungaricae, 1978, V.87, [8] Linden H., Applic. Anal., 1979, V.9, [9] Barkwell L., Lancaster P., Markus A.S., Can. J. Math., 1992, V.44, [10] Binding P., Applic. Anal., 1981, V.12, [11] Kostuchenko A.G., Shkalikov A.A., Funct. Anal. Appl., 1983, V.17, [12] Langer H., Int. J. Funct. Anal., 1973, V.12, [13] Markus A.S., Matsaev V.I., Math. USSR Sb., 1988, V.61, [14] Roach G.F., Sleeman B.D., Applic. Anal., 1979, V.9, [15] Zheng T.S., Liu W.M., Cai Z.B., Computers and Structures. An International Journal, 1989, V.33, [16] Goerisch F., Haunhorst H., J. Appl. Math. Mech. ZAMM, 1985, V.3,

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